A companion to the Nagaoka–Amari structure on the denormalized state space

Abstract This paper investigates Markov-invariant geometric structures on the denormalized state space, the space of strictly positive measures on a finite set. Within the subclass of structures compatible with the classical Nagaoka–Amari structure, we characterize all flat structures on the denormalized state space and identify a distinct one that complements the classical Nagaoka–Amari structure. This companion flat structure sheds new light on the $$\alpha =\pm 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> symmetry of the probability simplex and is conformally related to the Nagaoka–Amari structure, providing a coherent framework for understanding their relationship.